How do I connect vertices with a curve, then subdivide into more vertices that exactly match that curve?
You can apply a subdivision modifier to your mesh (You can also do that with a curve).



It will keep the same curvature.



You can also convert you mesh into a curve, change it the way you want, apply some subdivisions, and then reconvert that into a mesh.

Edit: You can increase the smooth value of the subdivision (on the left side, where you can set the number of subdivisions) or you can use the vertex smooth option in EditMode (on the same panel as before).


A.C